If you can't see anything simpler: when you have a rational function of sin and cos, you can always use the tangent half-angle substitution $t = \tan (x/2)$. Then $$\sin x = \frac{2t}{1+t^2}, \cos x = \frac{1-t^2}{1+t^2}, dx = \frac{2 dt}{1+t^2}.$$ Substituting these converts your integral into the integral of a rational function of $t$, which can then be integrated by partial fractions.